Peter D. Johnson, Lorenza Viola
We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global N-partite quantum state. Intuitively, we say that the multipartite state ''joins'' the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible N-partite quantum state exist is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic class of Werner states in d dimensions, and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as an extension of the N-representability problem to the case where the latter are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that Werner states enjoy, we determine in each case necessary and sufficient conditions for arbitrary d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement may generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. Implications for the joinability of arbitrary bipartite states as well as for quantum information processing tasks are discussed.
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http://arxiv.org/abs/1305.1342
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